This paper considers an unsourced random access (URA) setup equipped with a passive reconfigurable intelligent surface (RIS), where a massive number of unidentified users (of which only a small fraction are active at a given time) share the same communication resources. We propose a slotted transmission scheme that operates in two phases. In the first phase, called the RIS configuration phase, the base station (BS) detects the active pilots and estimates their respective channel state information (CSI). Then, using the estimated CSI, the BS suitably selects the phase shifts of the RIS elements. In the second phase, called the data phase, transmitted messages of active users are decoded. The proposed channel estimator offers the capability to estimate the channel coefficients of the users whose pilots interfere with each other without prior access to the list of selected pilots or the number of active users. In this paper, we consider the direct link between the users and the BS to be completely blocked, and show that employing RIS improves the performance of the URA system by creating additional links between the BS and the users. The effectiveness of the proposed algorithms is confirmed through computer simulations.